On subset sums of r-sets

AbstractA finite set of distinct integers is called an r-set if it contains at least r elements not divisible by q for each q⩾2. Let f(n,r) denote the maximum cardinality of an r-set A ⊂ {1,2,…,n} having no subset sum Σεiai(εi=0 or 1, aiϵA) equal to a power of two. In this paper estimates for f(n,r) are obtained. We prove that limr→∞αr=0, where αr=limn→∞f(n,r)n. This result verifies a conjecture of Erdős and Freiman (1990)